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In this paper we are interested in moments of the Minkowski question mark function . It appears that, to some extent, the results are analogous to results obtained for objects associated with Maass wave forms: period functions, -series, distributions. These objects can be naturally defined for as well. Various previous investigations of are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals which involve , define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects.
A Poincaré–Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler–Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler characteristics of tangency and exit-region orbifolds. As a corollary, we express the index sum of the vector field induced on the inertia orbifold to the Euler characteristics of the associated underlying topological spaces.
New metrics and distances for linear codes over the ring are defined, which generalize the Gray map, Lee weight, and Bachoc weight; and new bounds on distances are given. Two characterizations of self-dual codes over are determined in terms of linear codes over . An algorithm to produce such self-dual codes is also established.
We study the Fibonacci sequence mod for some positive integer . Such a sequence is necessarily periodic; we introduce a function which gives the ratio of the length of this period to itself. We compute in certain cases and provide bounds for it which depend on the nature of the prime divisors of .
Following a similar treatment of the Baumslag–Solitar group by Bahls, we modify a transformation developed by Magnus to linearly order the group given by the presentation . We demonstrate how this same method will fail to admit such a treatment of the groups , .
For an integer and a partition , we let be the multiset of hook lengths of which are divisible by . Then, define and to be the number of partitions of such that is even or odd, respectively. In a recent paper, Han generalized the Nekrasov–Okounkov formula to obtain a generating function for . We use this generating function to prove congruences for the coefficients .
KEYWORDS: difference equation, periodic convergence, boundedness character, unbounded solutions, periodic behavior of solutions of rational difference equations, nonlinear difference equations of order greater than one, global asymptotic stability, 39A10, 39A11
We resolve several conjectures regarding the boundedness character of the rational difference equation
We show that whenever parameters are nonnegative, , and , unbounded solutions exist for some choice of nonnegative initial conditions. We also partly resolve a conjecture regarding the boundedness character of the rational difference equation
We show that whenever , unbounded solutions exist for some choice of nonnegative initial conditions.